arXiv subjects

Jakub Waclawek

Publications and source records attributed to Jakub Waclawek.

5 recordsLinked to original sources

arXiv2026

Optimal fractional discrete Hardy inequalities on the half-line

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-Δ)^α$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $α>0$, we prove that the fractional Hardy inequality $$(-Δ)^α\geq\frac{4^αΓ^2(α+1/2)}π\frac{Γ(2\,\cdot\,-1)}{Γ(2\,\cdot\,-1+2α)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.

math.CA↗

arXiv2026

Optimal discrete $p$-Hardy-Rellich-Birman inequalities

We present a theory for constructing optimal lower bounds for the discrete half-line $p$-Laplacian of higher order $\ell\in\mathbb{N}$ and general $p>1$. The abstract framework introduces higher-order monotonicity and asymptotic constraints on a parameter sequence that determines optimal weights. As a concrete application, we specialize the parameter sequence to deduce new optimal discrete $p$-Hardy ($\ell=1$), $p$-Rellich ($\ell=2$), and $p$-Birman ($\ell\geq 3$) inequalities.

math.CA↗

arXiv2026

The $p$-Hardy-Rellich-Birman inequalities on the half-line

The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal.

math.CA↗

arXiv2025

A Herglotz-Nevanlinna function from the optimal discrete $p$-Hardy weight

It was recently proved by Fischer, Keller, and Pogorzelski in [Integr. Equ. Oper. Theory, 95(24), 2023] that the classical discrete $p$-Hardy inequality admits an improvement, and the optimal $p$-Hardy weight $ω_{p}$ was determined therein. We prove that $ω_{p}$ directly corresponds to a Herglotz-Nevanlinna function, establish an integral representation for this function, and consequently confirm a slight modification of a conjecture on its absolute monotonicity from the aforementioned article.

math.CA↗

arXiv2024

Optimal discrete Hardy-Rellich-Birman inequalities

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\ell=2$) and Birman ($\ell\geq3$) weights. For $\ell=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\ell=2$, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For $\ell\geq3$, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal.

math.CA↗